what is the minimum value that can take by the expression |5^(4m+3)-n^2| if m and n are non negative integers

Question

I tried everything I can.Please help me

 

Answer 1

Using solver, from excel I obtained the following answer:
Cell Name                                       Original Value                         Final Value
$H$4 absolute value of 54m + 3 - n2   124                                           4

To minimize the expression, the values are m= 0 and n=11 and the minimum value of the  expression is 4

Cell Name Original Value Final Value Integer
$H$5 m 0 0 Integer
$H$6 n 1 11 Integer

Cell Name Cell Value Formula Status Slack
$H$5 m 0 $H$5>=0 Binding 0
$H$6 n 11 $H$6>=0 Not Binding 11
$H$5=Integer
$H$6=Integer

Answer 2

absolute value of 5^{4m + 3} - n²

 

the smallest possible value for an absolute value expression is zero

 

the expression we have would equal zero if 5^{4m + 3} = n²

 

This is not possible because for all values of m, the power is an odd number. It is the even powers that can be perfect squares such as 25, 625, etc

 

So given zero is not a possibility, we need another strategy to make the value a minimum

 

when m = 0, 5^{4m + 3} = 125 and when n = 11, n² = 121, and the expression has a value of 4

 

This difference gets larger as m increases so the minimum value is 4